A-level · Beta

Solving trigonometric equations

Find every solution in a full turn, using a graph and a unit circle.

Move the target line. Select a solution to locate it on the graph and unit circle.

sin⁡(1x)=0.5,0≤x<2π\sin(1x)=0.5,\quad0\leq x<2\pi
Move the target line. Select a solution to locate it on the graph and unit circle.0-1.251.5708-0.6253.141604.71240.6256.28321.25x (radians)yT
The point uses angle b × x.angle b × x = 30°

Every red intersection is a solution; x = 2π is excluded.

Solutions in this interval
0.5236 rad (30°); 2.618 rad (150°)
Number of distinct solutions
2
Trace output
0.5

The buttons show rounded degree values. Inverse sine or cosine gives a principal value, not every solution. The green unit-circle point uses b × x.

Predict, test and explain

Step 1 of 3

Predict

How many solutions does sin x = ½ have for 0 ≤ x < 2π?

InvestigateWhy does inverse sine give only one of the two solutions?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

How many solutions does sin x = ½ have for 0 ≤ x < 2π?

sin⁡x=12\sin x=\tfrac12

Hint

Use symmetry and periodicity to find every solution in the stated interval.

Reveal answer

Both π/6 and 5π/6 give sine ½. Inverse sine returns only the principal value.

x=π6,5π6x=\frac\pi6,\frac{5\pi}6

Build an explanation

Hint 1

Use symmetry and periodicity to find every solution in the stated interval.

Hint 2

Both π/6 and 5π/6 give sine ½. Inverse sine returns only the principal value.

Hint 3

Compare the result with the model, then explain why it occurs.

Worked example

This example uses fixed values, separate from the diagram controls.

Solve sin(2x) = ½ for 0 ≤ x < 2π.

  1. For the angle 2x, use π/6 and 5π/6, then add 2π.
  2. Since 0 ≤ 2x < 4π, there are four angle solutions.
  3. Divide every angle by 2.

Answer: x = π/12, 5π/12, 13π/12, 17π/12.

Connect this idea

Watch out: Both π/6 and 5π/6 give sine ½. Inverse sine returns only the principal value.

Specification and learning route

AQA E7 · Edexcel Pure 5.7

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Trig graphs and periodicity; Exact values; Algebraic equations; Basic trig identities.

AQA specification · Edexcel specification